5.2 Spectral sequences and the chain complexes
5.2.1 For real functions
There are several different filtrations for a real Morse-Bott complex, we refer to [18], [3] and [16]. A straightforward filtration can be based on the value of each connected component under f , however, this approach requires more restrictions (the self-indexing condition) on the destination of trajectories in the closed 1-form situation. Therefore, we adopt a slightly more sophisticated approach which is consistent with the no homoclinic cycle condition in the closed 1-form case.
The ordering
Namely, we order the connected components according to the destination of the flow. Consider the set of connected components {Ci} of C as a set of vertices {vi}, each
vertex vi corresponds to one component Ci, and we connect two vertices vi, vj with
an directed edge eij(eji) if there are trajectories going from Ci to Cj (or vice versa),
the direction is towards where the trajectories point. Then we obtain a directed graph without loops. We perform the following algorithm to get an ordering:
Notice the number of components are finite due to the compactness of the manifold, so we do the following algorithm by finite repetition, in round j where j = 0, . . . , n:
1. For a vertex, check whether there are incident edges pointing out, label it j if there is not, otherwise repeat the check for the successive vertices where the edges are pointing to.
5.2. Spectral sequences and the chain complexes 66 2. Having exhausted all the vertices by the above operation, collect the labelled ones in a union C(j), delete them together with any incident edges from the graph, and start it over again as round j + 1.
We end up having an empty graph after finitely many rounds, say n, and each vertex is assigned uniquely to a specific order, for instance, C(j) is the union of the components with order j. Let us formalise this notation:
Notation 5.2.1 For each j, suppose we reindex all the connected components with order j as Cα with α ∈ A(j), where A(j) is the collection of all the new indices of
the connected components with order j. Then we denote C(j) as the union of all the connected components with order j according to the above algorithm:
C(j) = [
α∈A(j)
Cα.
Morse-Bott chain complex and its filtration(chain inclusion map)
Now we want to define the Morse-Bott complex and build a spectral sequence ac- cording to the hierarchy of the connected components, which is convergent to the homology groups of M .
A direct way of doing this is to define the chain complex as a direct sum of chain complexes of the critical manifolds and then specify the boundary maps, but to build on the previous chapters, we first approximate the original function f by a generic Morse function with respect to the ordering, and then define the chain complex with boundary maps from counting trajectories in a standard way.
Consider a connected component Ci of the critical manifold C in M , where f is
the Morse-Bott function on M , assign Ci a generic Morse function fi : Ci → R, i.e.
the gradient vi of fi with respect to some Riemannian metric is transverse, then we
have a Morse complex CMS
∗ (Ci, fi, vi) of fi on Ci as
C∗MS(Ci, fi, vi) =
M
p∈Crit (fi)
Z,
generated by the critical points of fi. Having done this to every connected compo-
nent Ci of C, we want to modify the original f out of these Morse functions fi’s,
5.2. Spectral sequences and the chain complexes 67 so that we can eventually pin down the trajectories from one critical component to another to specific critical points of fi’s as we did in the real Morse function case of
Chapter 1. Please see [2] and [18] for similar operations.
Firstly, choose a tubular neighbourhood N (Ci) of Ci, then if z ∈ Ci is a point in
this critical component, we write a point in N (Ci) as (z, x1, . . . , xλ, yλ+1, . . . , ym−dim(Ci)) =
(z, x, y) ∈ N (Ci), where m is the dimension of M and λ = ind (Ci) is the index of
the critical component Ci. Let fi+(z, x, y) = fi(z) − x2 + y2 : N (Ci) → R+, and
choose a bump function ρi : M → [0, 1] such that
ρi|Ci = 1 and ρi|M −N (Ci) = 0.
Then function Fi = f + ρifi+ creates no extra critical points other than the ones of
fi and f , for ρi with small partial derivatives
∂ρi ∂x: ∂Fi ∂x(z, x, y) = ∂f ∂x(z, x, y) + ρi ∂fi+ ∂x (z, x, y) + ∂ρi ∂xf + i (z, x, y) = ∂f ∂x(z, x, y) − 2ρix + ∂ρi ∂xf + i (z, x, y) < 0. where ∂f
∂x(z, x, y) < 0 near Ci and is chosen to be sufficiently small to guarantee overall negative value.
Moreover, if p ∈ Ci is a critical point of Fi, then its index indFi(p) = ind (Ci) +
indfi(p) is the sum of the index of p as a critical point of fi in Ci and the index of
Ci. And the Hessian of Fi at p has the matrix form as
d2Fi|p = d2f |p+ d2fi+|p = 0 0 + − + + − 0 0
Repeat this process to fi on Ci for each i, we obtain a slightly perturbed Morse
function F out of the original f and fi+’s as
F = f +
X
i
ρifi+.
Denote v and vi to be gradient vector fields of F and fi respectively with v close
to v. Now we construct a chain complex homotopic to the simplicial chain complex according to our relative Morse theory of Chapter 1:
5.2. Spectral sequences and the chain complexes 68 Definition 5.2.2 Let F be a Morse approximation of a Morse-Bott function f on
M , and v be the gradient vector field of F, then we define the Morse-Bott complex
of f as a direct sum of Morse chain complexes of the critical manifolds, and the boundary maps are derived from gradient v:
C∗MB(M, B, f ) = CMS(M, B, F, v) =
M
i
C∗−ind (CMS
i)(Ci, fi),
and the boundary map ∂λ : CλMB(M, B, f ) → Cλ−1(M, B, f ) is defined exactly the
same as in the Morse function case: ∂λ(p) =
X
q∈Critλ−1(F)
[p : q]q,
where [p : q] is the incidence coefficient by counting trajectories of −v from p to q
with given signs according to a prescribed orientation of Ws(p, v
) and Wu(q, v).
Now we want to show that we are able to construct F with care so that the
gradient v of F preserves the ordering according to our algorithm in the beginning
of the chapter.
The following lemma makes this possible:
Lemma 5.2.3 For each two connected components Ci, Cj ⊂ C of C such that
Wu(Ci, v) ∩ Ws(Cj, v) = ∅,
and there are no broken trajectories between them, then there exist open neighbour- hoods N (Ci) and N (Cj) so that
Wu(N (Ci), v) ∩ Ws(N (Cj), v) = ∅.
In other words, the lemma states that if there are no trajectories flowing from Ci
to Cj, then we can find some small neighbourhoods of Ci and Cj so that there are
no trajectories flowing from the neighbourhood of Ci to the one of Cj. Note that,
when i < j, according to our algorithm, it is automatically true that there exist no broken trajectories from Ci to Cj.
Proof : We prove it by contradiction. Firstly, let us assume f (Ci) < f (Cj),
otherwise, the gradient of f will guarantee no flow from the level set at Ci to the
level set at Cj.
5.2. Spectral sequences and the chain complexes 69 Suppose for the i, j such that Wu(Ci, v) ∩ Ws(Cj, v) = ∅, any open neighbour-
hoods N (Ci) and N (Cj) there always exist trajectories from N (Ci) to N (Cj):
Wu(N (Ci), v) ∩ Ws(N (Cj), v) 6= ∅.
Let {Nk(Ci)} and {Nl(Cj)} be sequences of nested neighbourhoods of Ci and
Cj, respectively, with Nk+1(Ci) ⊂ Nk(Ci) and Nl+1(Cj) ⊂ Nl(Cj), so that
Wu(Nk(Ci), v) ∩ Ws(Nl(Cj), v) 6= ∅, and \ k Nk(Ci) = Ci and \ l Nl(Cj) = Cj.
Now for each k = l, choose a trajectory γk from Nk(Ci) to Nk(Cj). We want to
show that such condition leads us to one of the two situations that contradict the hypothesis. Namely, the sequence {γk}k of trajectories between neighbourhoods of
Ci and Cj will converge to either a trajectory from Ci to Cj or a broken trajectory
that pass by some Cj0, where Cj0 is a critical manifold that lies between Ci and
Cj, meaning f (Ci) < f (Cj0) < f (Cj). (In fact, there can be more than one critical
manifold between Ci and Cj, but the argument is essentially the same, so we stick
to the simplest case. On the other hand, in the case of no such critical manifold in the middle, the argument can be simplified accordingly.)
Suppose Cj0 is such a connected critical manifold between Ci and Cj. Then
let {x1k} and {x2
k} be the two sequences of points lying on trajectories between
the nested neighbourhoods {Nk(Ci)} and {Nk(Cj)}. Moreover, we assume {x1k} is
between Ci and Cj0, and {x2
k} is between Cj0 and Cj. Since M − N0(Ci) ∪ N0(Cj)
is compact, the limits x1 and x2 are contained in M − N
0(Ci) ∪ N0(Cj), and we
claim the trajectory where x1 lies originates from C i.
Suppose the trajectory in which x1 lies does not come from C
i. Then we can
assume that there exists some k0 such that γx1 ∩ Nk0(Ci) = ∅. By the continuity
of the flow, there exists a neighbourhood N (x1) of x1 such that for any points
y ∈ N (x1) the trajectory γ
y ∩ Nk0(Ci) = ∅. Since x
1 is the limit of a subsequence
of {x1
k}, there exist xkn ∈ {xk} such that xkn ∈ N (x
1) also lie in N (x1) for each
kn > k0. In particular, this says xkn ∈ W/ u(N
kn(Ci), v) ∩ W
s(N (C
5.2. Spectral sequences and the chain complexes 70 N (Cj0), i.e. γx
nk does not belong to W u(N
nk(Ci), v), a contradiction to our choice
of the collection {γk}k. Therefore, x1 comes from Ci. Symmetrically, x2 ends in Cj.
Finally, consider the trajectory originates from Ci where x1 lies, it can either
ends in Cj0 first and then resumes there and reaches Cj after passing through x2;
or it bypasses Cj0 and reaches Cj directly. But either way, it contradicts to the
hypothesis in the statement. 2
Proposition 5.2.4 With the above notation, for each Cα ∈ A(j), there exists an
open neighbourhood N (Cα) of Cα so that the N (Cα) respects the ordering of the
algorithm, in other words, replace Cα by N (Cα), the algorithm will label N (Cα) by
j the same as Cα. 2
Then choose v sufficiently close to v, we obtain a filtration of C∗MB(M, B, f ) by
writing C∗(k)(F) = k M j=1 M α∈A(j) C∗MS(Cα, fα),
where fα : Cα → R is the Morse function that approximates f on Cα. Then
C∗(1)(F) ,→ C∗(2)(F) ,→ · · · ,→ C∗(n)(F) = C∗MB(M, B, f )
induces the desired spectral sequence with E1 term: Ek,l1 = M α∈A(k) Hk+l−ind (Cα)(Cα, fα) ∼= M α∈A(k) Hk+l−ind (Cα)(Cα).
We summarise the result in the following corollary:
Corollary 5.2.5 The homology of the Morse-Bott complex is isomorphic to the rel- ative homology of the underlying manifold, therefore, the spectral sequence induced by the filtered Morse-Bott complex converges to the relative homology:
Ek,lr ⇒ HMB
k+l(M, B, f ) ∼= Hk+l(M, B) when r → ∞,
with E1 term as: Ek,l1 ∼= M α∈A(k) Hk+l−ind (CMS α)(Cα, fα) ∼= M α∈A(k) Hk+l−ind (Cα)(Cα),
where fα : Cα → R is the Morse function that approximates f on Cα. 2
5.2. Spectral sequences and the chain complexes 71 Morse inequalities
Notation 5.2.6 Denote βn = rank (Hn(M, B)) and βn(C(k)) = rank (Hn(C(k)) =
P
α∈A(k)dim Hn(Cα).
Corollary 5.2.7 There is a polynomial R(t) with non-negative coefficients such that X k,l βk+l−ind (C(k))(C(k))tk+l= X n βntn+ (1 + t)R(t)
Proof : From Theorem 5.2.1 we know:
βn=
X
k+l=n
rank (Ek,l∞)
We want to show that there exists non-negative polynomial R(t) such that: X n X k+l=n tnrank (Ek,l1 ) =X n X k+l=n tnrank (Ek,l∞) + (1 + t)R(t).
Since the filtration is finite and the spectral sequence will reach stability after finite pages, and it can be reduced to show for each r,
X n X k+l=n tnrank (Ek,lr ) =X n X k+l=n tnrank (Ek,lr+1) + (1 + t)Rr(t),
where Rr(t) is a non-negative polynomial.
Denote Zk,lr = ker(dr : Ek,lr → E r k−r,l+r−1) and similarly, B r k,l = Im (dr : Ek,lr → Er k−r,l+r−1), then
rank (Ek,lr ) = rank (Zk,lr ) + rank (Bk,lr ), (5.1) and by the construction of the spectral sequence, Eklr+1 = Zklr/Bk+r,l−r+1r , therefore, rank (Ek,lr+1) = rank (Zk,lr ) − rank (Bk+r,l−r+1r ), (5.2) so
rank (Ek,lr ) = rank (Ek,lr+1) + rank (Bk,lr ) + rank (Bk+r,l−r+1r ), therefore X k+l=n rank (Ek,lr ) = X k+l=n rank (Ek,lr+1) + X k+l=n rank (Bk,lr ) + X k+l=n rank (Brk+r,l−r+1) (5.3)
5.2. Spectral sequences and the chain complexes 72 Now take the alternating sum of the above Equation (5.3) from n = 0 to n = N for any N , we have
n=N X n=0 X k+l=n (−1)N −nrank (Ek,lr ) = n=N X n=0 X k+l=n (−1)N −nrank (Ek.lr+1) + X p+q=k rank (Bp,qr ). This is equivalent to ∞ X n=0 X k+l=n tnrank (Ek,lr ) = ∞ X n=0 X k+l=n tnrank (Ek,lr+1) + (1 + t)Rr(t),
for some non-negative polynomial Rr(t).
Suppose the spectral sequence becomes stable at page r0, i.e., Er0 = E∞, then
after repeating this process r0 times, and taking the sum of them all, we have the
following: X n X k+l=n tnrank (Ek,l1 ) = X n X k+l=n tnrank (Er0 k,l) + (1 + t) r0−1 X r=1 Rr(t). Now R(t) =Pn−1
r=1Rr(t) is non-negative and the proof is complete. 2